Problem: $\dfrac{ -2v - 7w }{ 5 } = \dfrac{ 3v - 5x }{ -2 }$ Solve for $v$.
Multiply both sides by the left denominator. $\dfrac{ -2v - 7w }{ {5} } = \dfrac{ 3v - 5x }{ -2 }$ ${5} \cdot \dfrac{ -2v - 7w }{ {5} } = {5} \cdot \dfrac{ 3v - 5x }{ -2 }$ $-2v - 7w = {5} \cdot \dfrac { 3v - 5x }{ -2 }$ Multiply both sides by the right denominator. $-2v - 7w = 5 \cdot \dfrac{ 3v - 5x }{ -{2} }$ $-{2} \cdot \left( -2v - 7w \right) = -{2} \cdot 5 \cdot \dfrac{ 3v - 5x }{ -{2} }$ $-{2} \cdot \left( -2v - 7w \right) = 5 \cdot \left( 3v - 5x \right)$ Distribute both sides $-{2} \cdot \left( -2v - 7w \right) = {5} \cdot \left( 3v - 5x \right)$ ${4}v + {14}w = {15}v - {25}x$ Combine $v$ terms on the left. ${4v} + 14w = {15v} - 25x$ $-{11v} + 14w = -25x$ Move the $w$ term to the right. $-11v + {14w} = -25x$ $-11v = -25x - {14w}$ Isolate $v$ by dividing both sides by its coefficient. $-{11}v = -25x - 14w$ $v = \dfrac{ -25x - 14w }{ -{11} }$ Swap signs so the denominator isn't negative. $v = \dfrac{ {25}x + {14}w }{ {11} }$